Simplifying Rational Expressions A Step By Step Guide

Jul 12, 2025 by ADMIN 54 views

This comprehensive guide will walk you through the process of simplifying the rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ . We will explore the necessary steps, including factoring polynomials, identifying common factors, and canceling them out to arrive at the simplest form of the expression. Understanding how to simplify rational expressions is a fundamental skill in algebra and calculus, enabling you to solve more complex equations and problems. So, let's delve into the process and simplify this expression together.

Understanding Rational Expressions

Rational expressions, at their core, are fractions where the numerator and the denominator are polynomials. Just like numerical fractions, rational expressions can be simplified to their lowest terms. Simplifying rational expressions is a crucial skill in algebra as it allows us to work with expressions in their most manageable form, making it easier to perform operations like addition, subtraction, multiplication, and division. It also plays a significant role in solving equations and understanding the behavior of functions.

The process of simplifying rational expressions involves factoring both the numerator and the denominator, identifying common factors, and then canceling those common factors. This is analogous to simplifying numerical fractions, where we find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. For example, the fraction 6/8 can be simplified to 3/4 by dividing both the numerator and the denominator by their GCD, which is 2. Similarly, in rational expressions, we factor the polynomials and look for common polynomial factors to cancel out. Understanding the underlying principles of factoring and cancellation is key to mastering the simplification of rational expressions. The ability to manipulate these expressions efficiently is not only essential for success in algebra but also in higher-level mathematics courses like calculus and differential equations.

Step 1: Factoring the Numerator

Our first crucial step in simplifying the rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ is to factor the numerator, which is the quadratic expression $z^2 + 12z + 32$ . Factoring a quadratic expression involves rewriting it as a product of two binomials. To achieve this, we need to identify two numbers that satisfy two specific conditions: their product must equal the constant term (32 in this case), and their sum must equal the coefficient of the linear term (12 in this case). Let's break down the process of finding these numbers.

We begin by listing pairs of factors of 32: 1 and 32, 2 and 16, 4 and 8. Now, we examine each pair to see which one adds up to 12. We can quickly see that the pair 4 and 8 fits the criteria because 4 multiplied by 8 equals 32, and 4 plus 8 equals 12. With these numbers in hand, we can rewrite the quadratic expression as a product of two binomials. The factored form of $z^2 + 12z + 32$ is $(z + 4)(z + 8)$ . This means that if we were to multiply these two binomials together using the distributive property (also known as FOIL), we would arrive back at the original quadratic expression. Factoring the numerator is a fundamental step because it allows us to identify potential common factors with the denominator, which we will address in the next step. This process of factoring quadratic expressions is a core skill in algebra, and mastering it will greatly aid in simplifying rational expressions and solving related problems.

Step 2: Factoring the Denominator

Having successfully factored the numerator, our next essential task is to factor the denominator of the rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ , which is the cubic expression $z^3 + 4z^2 - 32z$ . Factoring this cubic expression requires a slightly different approach compared to factoring the quadratic numerator. The first step in factoring any polynomial is to look for a greatest common factor (GCF) among all the terms. In this case, we can observe that each term in the expression has a factor of 'z'. Therefore, we can factor out 'z' from the entire expression.

Factoring out 'z' from $z^3 + 4z^2 - 32z$ gives us $z(z^2 + 4z - 32)$ . Now, we are left with a quadratic expression inside the parentheses, which we can factor using the same method we applied to the numerator. We need to find two numbers that multiply to -32 and add up to 4. Let's consider the factor pairs of 32: 1 and 32, 2 and 16, 4 and 8. Since we need a product of -32, one of the numbers must be negative. By trying different combinations, we find that the numbers 8 and -4 satisfy our conditions because 8 multiplied by -4 equals -32, and 8 plus -4 equals 4. Thus, we can factor the quadratic expression $z^2 + 4z - 32$ as $(z + 8)(z - 4)$ .

Putting it all together, the factored form of the denominator $z^3 + 4z^2 - 32z$ is $z(z + 8)(z - 4)$ . This complete factorization of the denominator is crucial because it allows us to identify any factors that are common with the factored numerator. Identifying these common factors is the key to simplifying the rational expression, as we will see in the next step. Mastering the technique of factoring out the GCF and then factoring the remaining quadratic expression is a fundamental skill in simplifying polynomial expressions and solving related algebraic problems.

Step 3: Identifying Common Factors

Now that we have successfully factored both the numerator and the denominator of the rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ , we can move on to the crucial step of identifying common factors. This step is essential because it allows us to simplify the expression by canceling out these common factors. We have the numerator factored as $(z + 4)(z + 8)$ and the denominator factored as $z(z + 8)(z - 4)$ .

By carefully comparing the factored forms of the numerator and the denominator, we can easily spot the common factor. In this case, we observe that the binomial $(z + 8)$ appears in both the numerator and the denominator. This means that $(z + 8)$ is a common factor, and we can proceed to the next step where we will cancel out this common factor. Identifying common factors is a fundamental skill in simplifying rational expressions and working with fractions in general. It's analogous to simplifying numerical fractions by dividing both the numerator and the denominator by their greatest common divisor. For instance, in the fraction 10/15, the common factor is 5, and dividing both the numerator and denominator by 5 simplifies the fraction to 2/3. Similarly, in rational expressions, identifying and canceling common factors helps us reduce the expression to its simplest form, making it easier to work with in subsequent algebraic manipulations or calculations.

Step 4: Canceling Common Factors

Having identified the common factor of $(z + 8)$ in both the numerator and the denominator of our rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ , we can now proceed to cancel it out. Canceling common factors is a fundamental step in simplifying rational expressions, similar to reducing numerical fractions to their lowest terms. This process involves dividing both the numerator and the denominator by the common factor.

In our case, we have the expression with the numerator and denominator factored as $\frac{(z + 4)(z + 8)}{z(z + 8)(z - 4)}$ . To cancel the common factor $(z + 8)$ , we divide both the numerator and the denominator by $(z + 8)$ . This effectively removes the factor $(z + 8)$ from both the numerator and the denominator, leaving us with a simplified expression. After canceling the common factor, the expression becomes $\frac{(z + 4)}{z(z - 4)}$ .

The act of canceling common factors is based on the principle that dividing a quantity by itself equals 1, as long as the quantity is not zero. In other words, $\frac{(z + 8)}{(z + 8)} = 1$ , provided that $z \neq -8$ . This condition is important because dividing by zero is undefined in mathematics. Therefore, when we cancel the common factor, we are essentially multiplying the expression by 1, which does not change its value but simplifies its form. The ability to confidently cancel common factors is a key skill in simplifying rational expressions and is essential for working with algebraic fractions effectively.

Step 5: Stating Restrictions (Important!)

While canceling common factors simplifies the rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ , it's crucial to remember that this simplification comes with a caveat: we must state the restrictions on the variable. Restrictions arise from the original expression's denominator, as division by zero is undefined in mathematics. In other words, any value of the variable that would make the denominator equal to zero must be excluded from the domain of the expression.

To determine the restrictions, we look at the factored form of the original denominator, which was $z(z + 8)(z - 4)$ . We need to identify the values of 'z' that would make any of these factors equal to zero. Setting each factor equal to zero, we get the following equations:

z = 0
z + 8 = 0
z - 4 = 0

Solving these equations, we find the following restrictions: z = 0, z = -8, and z = 4. These are the values of 'z' that would make the original denominator zero, and therefore, they must be excluded from the domain of the simplified expression. Stating the restrictions is a vital part of simplifying rational expressions because it ensures that the simplified expression is equivalent to the original expression for all permissible values of the variable. Failing to state the restrictions would mean that the simplified expression might be considered defined for values where the original expression was undefined, leading to potential errors in calculations or problem-solving. Therefore, always remember to identify and state the restrictions after simplifying a rational expression.

Final Simplified Expression

After factoring, canceling common factors, and stating the restrictions, we have arrived at the final simplified form of the rational expression $\frac{z^2+12 z+32}{z^3+4 z^2-32 z}$ . The simplified expression is $\frac{z + 4}{z(z - 4)}$ , with the restrictions that z cannot be equal to 0, -8, or 4. This means that the simplified expression is equivalent to the original expression for all values of z except for these three.

The process we followed involved several key steps: first, we factored both the numerator and the denominator to identify common factors. Then, we canceled out the common factor of (z + 8) from both the numerator and the denominator. Finally, we determined and stated the restrictions on the variable z, which are z ≠ 0, z ≠ -8, and z ≠ 4. These restrictions are crucial because they ensure that the simplified expression is a true representation of the original expression, excluding values that would make the original expression undefined due to division by zero.

The simplified expression $\frac{z + 4}{z(z - 4)}$ is now in its lowest terms, meaning that there are no more common factors that can be canceled out. This simplified form is easier to work with in further algebraic manipulations, such as solving equations or performing operations like addition or subtraction with other rational expressions. The ability to simplify rational expressions is a fundamental skill in algebra and higher-level mathematics, and mastering this process is essential for success in these fields.

Therefore, the final answer, including both the simplified expression and the restrictions, is:

$\frac{z + 4}{z(z - 4)}$ , where $z \neq 0$ , $z \neq -8$ , and $z \neq 4$ .

This result represents the most concise and accurate form of the original rational expression, taking into account all necessary conditions and limitations.